What Is Nonlocal in Counterfactual Quantum Communication

We revisit the ``counterfactual quantum communication'' of Salih et al. [1], who claim that an observer ``Bob'' can send one bit of information to a second observer ``Alice'' without any physical particle traveling between them. We show that a locally conserved, massless current---specifically, a current of modular angular momentum, ${L}_{z}$ mod $2\ensuremath{\hbar}$---carries the one bit of information. We integrate the flux of ${L}_{z}$ mod $2\ensuremath{\hbar}$ from Bob to Alice and show that it equals one of the two eigenvalues of ${L}_{z}$ mod $2\ensuremath{\hbar}$, either 0 or $\ensuremath{\hbar}$, thus precisely accounting for the one bit of information he sends her. We previously [2] obtained this result using weak values of ${L}_{z}$ mod $\ensuremath{\hbar}$; here we do not use weak values.